[mostanad]

Hi Everyone,
I hope you are all well. I have a question about the suitable equation for my field.
You can see the following image showing a field named nu.





As you see, I need to diffuse (interpolate) the values from the blue region to the red region, in order to smooth the values. The direction for this diffusion is shown by the yellow arrows. I have the set of cells containing the high and low nu values, separating them.

However, I need to know what is the suitable way to have an smooth diffusion from the blue region to red region?


Sincerely yours,
Mohammad

[FMDenaro]

Use a linear interpolation over triangles, in a FE manner

[mostanad]

Originally Posted by FMDenaro

Use a linear interpolation over triangles, in a FE manner

Thank you Filippo,
Can you introduce me a refrence for that?
Thanks a lot.

[FMDenaro]

Originally Posted by mostanad

Thank you Filippo,
Can you introduce me a refrence for that?
Thanks a lot.

However, your interpolation requires only linear (if triangle) or bilinear lagrangian interpolation, it is very easy to do.



The fundamental textbook about FE is by Olek C Zienkiewicz

[mostanad]

Originally Posted by FMDenaro

However, your interpolation requires only linear (if triangle) or bilinear lagrangian interpolation, it is very easy to do.



The fundamental textbook about FE is by Olek C Zienkiewicz

Thanks. But can you tell me it is something like Laplacian equation? I need to know what is the idea to diffuse the values? based on distance?

[FMDenaro]

Originally Posted by mostanad

Thanks. But can you tell me it is something like Laplacian equation? I need to know what is the idea to diffuse the values? based on distance?

This is a simple 1D example of linear interpolation effects:

f_int= 0.5*(f1 + f2)

You can assume

f_exact= f_int + truncation

truncation = - 0.5*(f1 + f2) + f_exact
= - 0.5*h^2 * ( f1 - 2*f_exact + f2)/h^2

So that you see the appearence of a smoothing terms in the form of second derivatives

[mostanad]

Originally Posted by FMDenaro

This is a simple 1D example of linear interpolation effects:

f_int= 0.5*(f1 + f2)

You can assume

f_exact= f_int + truncation

truncation = - 0.5*(f1 + f2) + f_exact
= - 0.5*h^2 * ( f1 - 2*f_exact + f2)/h^2

So that you see the appearence of a smoothing terms in the form of second derivatives

So it is something like averaging method? How do you relate it to the neighboring cells? I can't find your point, as I need to have the values over the cells affected by the known values. In my opinion, it should be something like spreading over the cells using Laplacian equation. Then, having zero gradient BC can be a good option for this equation. Am I right?

[FMDenaro]

Originally Posted by mostanad

So it is something like averaging method? How do you relate it to the neighboring cells? I can't find your point, as I need to have the values over the cells affected by the known values. In my opinion, it should be something like spreading over the cells using Laplacian equation. Then, having zero gradient BC can be a good option for this equation. Am I right?

This an equivalent of the differential smooth filtering. You can just substitute the nodal value you compute by means of the resulting linearly interpolated value, resulting in a spreading.

[mostanad]

Originally Posted by FMDenaro

This an equivalent of the differential smooth filtering. You can just substitute the nodal value you compute by means of the resulting linearly interpolated value, resulting in a spreading.

Yes, you are right but I require a different thing.
The thing I need is to have a diffusive equation over the red domain, with giving the values at boundaries with zero gradient.
The operation you mentioned is just an interpolation over the unknown cells and does not definitely satisfy the zero-gradient condition.


Please help me with this requirement.

[FMDenaro]

Originally Posted by mostanad

Yes, you are right but I require a different thing.
The thing I need is to have a diffusive equation over the red domain, with giving the values at boundaries with zero gradient.
The operation you mentioned is just an interpolation over the unknown cells and does not definitely satisfy the zero-gradient condition.


Please help me with this requirement.

This is not your original question. If you set an elliptic equation in the red region with homogeneus Neumann BC. on the circle, you have infinite solution. You can fix an arbitrary value.

[mostanad]

Originally Posted by FMDenaro

This is not your original question. If you set an elliptic equation in the red region with homogeneus Neumann BC. on the circle, you have infinite solution. You can fix an arbitrary value.

OK Filippo,
Does high viscosity, like the above image, mean to be like an obstacle for the flow? Does it prevent the fluid to enter the higher viscosity domain?
I can't understand what happens if the viscosity is high and low between two subdomains.

[FMDenaro]

Originally Posted by mostanad

OK Filippo,
Does high viscosity, like the above image, mean to be like an obstacle for the flow? Does it prevent the fluid to enter the higher viscosity domain?
I can't understand what happens if the viscosity is high and low between two subdomains.

No. The continuity constraint requires that V.n=0 on the surface of the high viscosity region. This is a condition on the mass flux and has nothing to do with the diffusion. Thus, you have to compute and check if that condition is fulfilled.

[mostanad]

Originally Posted by FMDenaro

No. The continuity constraint requires that V.n=0 on the surface of the high viscosity region. This is a condition on the mass flux and has nothing to do with the diffusion. Thus, you have to compute and check if that condition is fulfilled.

OK. But I only need to know what will be the effect if the viscosity over a set of cells will be high?


You can see in the following video, when the viscosity is high in block, the flow is harder to penetrate.
https://www.youtube.com/watch?v=ueMCheEpMrk

[FMDenaro]

Originally Posted by mostanad

OK. But I only need to know what will be the effect if the viscosity over a set of cells will be high?


You can see in the following video, when the viscosity is high in block, the flow is harder to penetrate.
https://www.youtube.com/watch?v=ueMCheEpMrk

Showing coloured movies means nothing. "Harder to penetrate" means also nothing ... You need to explicitly compute the continuity condition to check what happens. Also high viscosity flows (Stokes flows) remain coupled with the divergence-free constraint that you have to satisfy. High viscosity does not mean automatically a region that cannot be penetrated by a flux of mass.

Consider a closed surface for the red region and evaluate your error in the continuity. Then refine the grid and check if the error decreases accordingly. This is a quantitative way to evaluate if your method works.

[mostanad]

Originally Posted by FMDenaro

Showing coloured movies means nothing. "Harder to penetrate" means also nothing ... You need to explicitly compute the continuity condition to check what happens. Also high viscosity flows (Stokes flows) remain coupled with the divergence-free constraint that you have to satisfy. High viscosity does not mean automatically a region that cannot be penetrated by a flux of mass.

Consider a closed surface for the red region and evaluate your error in the continuity. Then refine the grid and check if the error decreases accordingly. This is a quantitative way to evaluate if your method works.

No. You always evade from my main question. I told you what will be the flow cpndition if we only prescribe a high viscosity to some limited cells. I need only the answer for this.

[FMDenaro]

Originally Posted by mostanad

No. You always evade from my main question. I told you what will be the flow cpndition if we only prescribe a high viscosity to some limited cells. I need only the answer for this.

If you didn't focus your problem I cannot help. You cannot have an aswer if you are not aware that Stokes flows are governed by a system of equations. Setting high viscosity in the momentum equations in a certain region does not satisfies automatically the continuity equation in that region (and along its boundary).
Instead of posting colours, compute the mass equation to check if your method works. Compute the Div v field in the whole space, cell-by cell. Then plot v.n along the circle over the red region.

[mostanad]

Originally Posted by FMDenaro

If you didn't focus your problem I cannot help. You cannot have an aswer if you are not aware that Stokes flows are governed by a system of equations. Setting high viscosity in the momentum equations in a certain region does not satisfies automatically the continuity equation in that region (and along its boundary).
Instead of posting colours, compute the mass equation to check if your method works. Compute the Div v field in the whole space, cell-by cell. Then plot v.n along the circle over the red region.

Am I asking about no penetration condition? Not now. So my main question is what is the effect of havong high viscosity in a specific region of domain? These are not my results. I am looking for someone who has the cfd understanding about different domains with different properties.

[FMDenaro]

“ Does high viscosity, like the above image, mean to be like an obstacle for the flow? Does it prevent the fluid to enter the higher viscosity domain?
I can't understand what happens if the viscosity is high and low between two subdomains.”


Ask to your CFD expert but before that make your brain clearer when you post a question.

[mostanad]

Originally Posted by FMDenaro

“ Does high viscosity, like the above image, mean to be like an obstacle for the flow? Does it prevent the fluid to enter the higher viscosity domain?
I can't understand what happens if the viscosity is high and low between two subdomains.”


Ask to your CFD expert but before that make your brain clearer when you post a question.

"Not now". Read it again.